Certification Problem

Input (COPS 720)

We consider the TRS containing the following rules:

b	→	f(f(b))	(1)
f(a)	→	c	(2)
f(a)	→	b	(3)
b	→	f(h(a,b))	(4)

The underlying signature is as follows:

{b/0, f/1, a/0, c/0, h/2}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2023)

1 Non-Joinable Fork

The system is not confluent due to the following forking derivations.

t₀	=	f(a)
	→	b
	=	t₁

t₀	=	f(a)
	→	c
	=	t₁

The two resulting terms cannot be joined for the following reason:

The reachable terms of these two terms are approximated via the following two tree automata, and the tree automata have an empty intersection.
- Automaton 1
  - final states:
    
    {8}
  - transitions:
    
    a → 48
    
    h(48,8) → 22
    
    f(22) → 8
    
    f(8) → 22
    
    b → 8
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {9}
  - transitions:
    
    c → 9
  The automaton is closed under rewriting as it is compatible.